1. Digital Systems: Logic Gates and Boolean Algebra
Wen-Hung Liao, Ph.D.

2. Objectives Perform the three basic logic operations.
Describe the operation of and construct the truth tables for the AND, NAND, OR, and NOR gates, and the NOT (INVERTER) circuit.
Draw timing diagrams for the various logic-circuit gates.
Write the Boolean expression for the logic gates and combinations of logic gates.
Implement logic circuits using basic AND, OR, and NOT gates.
Appreciate the potential of Boolean algebra to simplify complex logic circuits.

3. Objectives (cont’d)
Use DeMorgan's theorems to simplify logic expressions.
Use either of the universal gates (NAND or NOR) to implement a circuit represented by a Boolean expression.
Explain the advantages of constructing a logic-circuit diagram using the alternate gate symbols versus the standard logic-gate symbols.
Describe the concept of active-LOW and active-HIGH logic symbols.
Draw and interpret the IEEE/ANSI standard logic-gate symbols.

4. Boolean Constants and Variables
Boolean 0 and 1 do not represent actual numbers but instead represent the state, or logic level.
Logic 0
Logic 1
False
True
Off On
Low
High No
Yes
Open switch
Closed switch

5. Three Basic Logic OperationsOR
AND
NOT

6. Truth Tables
A truth table is a means for describing how a logic circuit’s output depends on the logic levels present at the circuit’s inputs.
Inputs
Output A B x 1 A ? x B

7. OR Operation Boolean expression for the OR operation: x =A + B
The above expression is read as “x equals A OR B”
Figure 3-2

8. OR Gate
An OR gate is a gate that has two or more inputs and whose output is equal to the OR combination of the inputs.
Figure 3-3

9. Example 3-1
Using an OR gate in an alarm system(refer to Fg03-04.ckt)

10. Example 3.2
Timing diagram (refer to Fg03-05.ckt)

11. AND Operation Boolean expression for the AND operation: x =A B
The above expression is read as “x equals A AND B”

12. AND Gate
An AND gate is a gate that has two or more inputs and whose output is equal to the AND product of the inputs.
Figure 3-8

13. Timing Diagram for AND Gate

14. Timing Diagram for AND Gate

15. Enable/Disable (inhibit) Circuit

16. Enable/Disable Circuit

17. NOT Operation
The NOT operation is an unary operation, taking only one input variable.
Boolean expression for the NOT operation: x = A
The above expression is read as “x equals the inverse of A”
Also known as inversion or complementation.
Can also be expressed as: A’
Figure 3-11 A

18. NOT Circuit Also known as inverter. Always take a single input
Application:

19. Describing Logic Circuits Algebraically
Any logic circuits can be built from the three basic building blocks: OR, AND, NOT
Example 1: x = A B + C
Example 2: x = (A+B)C
Example 3: x = (A+B)
Example 4: x = ABC(A+D)

20. Examples 1,2

21. Examples 3

22. Example 4

23. Evaluating Logic-Circuit Outputs
x = ABC(A+D)
Determine the output x given A=0, B=1, C=1, D=1.
Can also determine output level from a diagram

24. Figure 3.16

25. Implementing Circuits from Boolean Expressions
We are not considering how to simplify the circuit in this chapter.
y = AC+BC’+A’BC
x = AB+B’C
x=(A+B)(B’+C)

26. Figure 3.17

27. Figure 3.18

28. NOR Gate Boolean expression for the NOR operation: x = A + B
Figure 3-20: timing diagram

29. Figure 3.20

30. NAND Gate Boolean expression for the NAND operation: x = A B
Figure 3-23: timing diagram

31. Figure 3.23

32. Boolean Theorems (Single-Variable)
x* 0 =0
x* 1 =x
x*x=x
x*x’=0
x+0=x
x+1=1
x+x=x
x+x’=1

33. Boolean Theorems (Multivariable)
x+y = y+x
x*y = y*x
x+(y+z) = (x+y)+z=x+y+z
x(yz)=(xy)z=xyz
x(y+z)=xy+xz
(w+x)(y+z)=wy+xy+wz+xz
x+xy=x
x+x’y=x+y
x’+xy=x’+y

34. DeMorgan’s Theorems (x+y)’=x’y’
Implications and alternative symbol for NOR function (Figure 3-26)
(xy)’=x’+y’
Implications and alternative symbol for NAND function (Figure 3-27)
Example 3-17: Figure 3-28
Extension to N variables

35. Figure 3.26

36. Figure 3.27

37. Universality of NAND Gates

38. Universality of NOR Gates

39. Available ICs

40. Alternate Logic Symbols
Step 1: Invert each input and output of the standard symbol
Change the operation symbol from AND to OR, or from OR to AND.
Examples: AND, OR, NAND, NOR, INV

41. Alternate Logic-Gate Representation

42. Logic Symbol Interpretation
When an input or output on a logic circuit symbol has no bubble on it, that line is said to be active-HIGH.
Otherwise the line is said to be active-LOW.

43. Figure 3.34

44. Figure 3.35

45. Which Gate Representation to Use?
If the circuit is being used to cause some action when output goes to the 1 state, then use active-HIGH representation.
If the circuit is being used to cause some action when output goes to the 0 state, then use active-LOW representation.
Bubble placement: choose gate symbols so that bubble outputs are connected to bubble inputs, and vice versa.

46. Figure 3.36

47. IEEE Standard Logic Symbols
NOT
AND OR
NAND
NOR 1 A x A
& x B A ≧1 x A
& x A ≧1 x B B B